Mixing cell models

LDPE tabular reactor is divided into several reaction zon acoirding to fhe feed injection points. Here we apply mixing cell model for tobidar rcsictor which considea s the reactor axis as series of cells which is conceptually the same as CSTRs in series. In tiiis study 40 cells are used for each reactor spool of 10 m long. The mass balant equation of a single cell at steady state can be written as follows. 

This equation is predicted by the mixing cell model, and turbulence theories put forward by Aris and Amundson130 and by Prausnttz(31). 

A two-dimensional mixing-cell model has been constructed to simulate the distribution of temperature, concentration and current density in cross-flow monoliths. Preliminary model calculations are presented here. 

There is an obvious similarity between the equations of the crossflow model and those of the modified mixing-cell model. With suitable redefinition of parameters, it can easily be shown that these partial differential equations are mathematically identical. Thus, the solution for the modified mixing-cell model is identical to the solution for the crossflow model, for the same set of boundary conditions. 

Three-parameter PDE model (Van Swaaij et aL106) This model is largely used to correlate the RTD curves from a trickle-bed reactor. The model is based on the same concept as the crossflow or modified mixing-cell model, except that axial dispersion in the mobile phase is also considered. The model, therefore, contains three arbitrary parameters, two of which are the same as those used in the cross-flow model and the third one is the axial dispersion coefficient (or the Peclet number in dimensionless form) in the mobile phase (see Fig. 3-11). 

Three-parameter mixing-cell model (Van Swaaij et aL106) This is a stagewise model for liquid-phase backmixing in a trickle-bed reactor. According to this model, an elementary mixing pattern for trickle flow is expressed as106... 

Well-Mixed Cell Model. A conceptually simple approach is based on the representation of the airshed by a three-dimensional array of well-mixed vessels (34, 35, 36). As before, we assume that the airshed has been divided into an array of L cells. Instead of using the array simply as a tool in the finite-difference solution of the continuity equations, let us now assume that each of these cells is actually a well-mixed reactor with inflows and outflows between adjacent cells. If we neglect diffusive transport across the boundaries of the cells and consider only convective transport among cells, a mass balance on species i in cell k is given by... 

Therefore, the well-mixed cell model can also be described as the result of the finite difference approximation of the spatial derivatives of (7)— i.e., of the conservation equations in which diffusion has been neglected. 

MacCracken et al. (36) have applied the well-mixed cell model in describing pollutant transport and dispersion in the San Francisco Bay Area. 

Appelo C. A. J. and Willemsen A. (1987) Geochemical calculations and observations on salt water intrusions, a combine geochemical/mixing cell model. J. Hydrol. 94,... 

FIGURE 9.25 Mixing cell model (from Ranade, 1998). 

In Chapter 4 it was pointed out that the performance of a CSTR sequence approached that of a single PFR of equivalent total residence time as the number of units in a sequence approached infinity. This result is also obeyed by the F 6) and E 6) curves computed from the mixing-cell model reported in Figure 5.3. Since the plug-flow model represents one limit of the dispersion model (that when D 0), it is reasonable to assume that there is an interrelationship between mixing-cell and dispersion models that can be set forth for the more general case of finite values... 

Thus, the correct axial Peclet number for defining the equivalent length of a perfect mixing cell is 2. This, in turn, provides the relationship between the number of mixing-cell parameters of the mixing-cell model and the corresponding axial dispersion parameters of the dispersion model. 

A similar comparison may be made between the dispersion model and the forward/reverse flow mixing-cell model of Figure 5.4. This is perhaps even a physically more meaningful comparison, although we were unable to use the model to derive analytical expressions for the age distributions, because the forward/back-ward communication provided is more similar to the physical nature of the diffusion process. In this case the expression corresponding to equation (5-37) is... 

There is little further to be said concerning the application of this method. One interesting approach, however, that seems not to have been explored very much, is incorporation of the mixing-cell model representation for E t) in equation (5-1), as shown in Section 5.2.2. 

Mixing-cell models were discussed extensively in Chapter 4 under the guise of the analysis of CSTR sequences. It is a good time to revisit some of this analysis from the specific point of view of modeling nonideal reactors. 

Figure 5.14 (b) Effect of nonideal exit-age distribution on Type III selectivity in a tubular flow reactor (mixing-cell model). 

A Two-dimensional Mixing-Cell Model for Nonisothermal Reactors... 

Since nonisothermality in tubular reactors often leads to radial as well as axial gradients, and since the mixing-cell model in its one-dimensional form is not very convenient anyway, it seems logical to see what a two-dimensional mixing-cell model might entail. 

The entire mixing-cell model can then be represented by the following simultaneous equations... 

Figure 6.5 Results of a typical steady-state simulation with the mixing-cell model of Deans and Lapidus. [After H.A. Deans and L. Lapidus, Amer. Inst. Chem. Eng. J., 6, 663, with permission of the American Institute of Chemical Engineers, (I960).]...

Ikeda (1979/ 2> Langmuir Dubinin Pore diff. Thermal Equilib. CO2-5A Sieve Mixing cell model. Finite difference. 

The approximation of the river as a series of discrete well-mixed cells introduces additional dispersion into the model. Even if a value of = 0 is input, some dispersion wUl still be predicted by the model. Banks (1974) developed a mixed cell model which may be used to quantify this numerical dispersion ... 

Banks, R. B. 1974. A Mixing Cell Model for Longitudinal Dispersion in Open Channels, Water Resources Research, vol. 10, pp. 357-358. 

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